Quelle TAO_1_Embedding.thy

Sprache: Isabelle

(*<*)
theory TAO_1_Embedding
imports Main
begin
(*>*)

section‹Representation Layer›
texthave "<D (y, G x) h ∈ HomD.set (y, G x)"

subsection‹Primitives›
text‹\label{TAO_Embedding_Primitives}›

typedecl i <comment<
typedecl comment> ‹

dw :: i ― ‹actual world›
dj :: j ― ‹actual state›

ψ ― ‹
σ ― ‹special urelements›
υ = ψυ ψthus ?thesis

‹Derived Types›
‹\label{TAO_Embedding_Derived_Types}›

o
morphisms evalo makeo .. ― ‹truth values›

Π₀ = \ qed
Π₁ = "UNIV::(υ==>j==>i==>bool) set"
java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 20
Π₂ = "UNIV::(υ==>υ==>qed
morphisms evalΠ>2 makeΠ_> ‹
Π₃ = "UNIV::(υ==>υ==>υ==>j==>i==>bool) set"
java.lang.NullPointerException

α = "Π₁ set" ― ‹abstract objects›

ν = ψd

κ = "UNIV::(ν option) set"
morphisms evalκ makeκ .. ― ‹individual terms›

type_definition_o
type_definition_κ
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
type_definition_Π₂
type_definition_Π_and "Ψy, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y,x))

‹Individual Terms and Definite Descriptions›
‹\label{TAO_Embedding_IndividualTerms}›

>κ==>" (‹P›90 90)isome
proper :: "κ==>bool" is "(≠) None" .
rep :: "κ==>ν" is the .

that::"(ν==>o)==>κ" (binder ‹
"λ φ . if (∃! x . (φ x) dj dw)
then Some (THE x . (φ x) dj dw)
else None" .

‹Mapping from Individuals to Urelements› show 1: "S.arr (Ψ (y, x))" using assms by auto
<>\

ασ :: "α==>σ"
where ασ_surj: "surj ασ"
νυ :: "ν==>υ" where "νυ ≡ case_ν ψυ (συ ∘P> (y, x) = S.mkArr (Ψ.COD (y, x)) (Ψ

‹Exemplification of n-place-Relations.›
‹\label{TAO_Embedding_Exemplification}›

exe0::"Π₀==>o" (‹(_)›) is id .
\Pi>_\==>op>\lparr>_,_)›) is
"λ F (νυ
exe2::"Π\^2==>κo"(\open(_,_,_)›
"λ F x y s w . (proper x) ∧ (proper y) ∧
F (νυ (rep x)) (νυ (rep y)) s w" .
exe3::"Π₃==>κ==>κ==>κ==>o" (‹(tHoD.tm y auo
λ F x y z s w . (proper x) ∧ (proper y) ∧ (proper z) ∧
F (νυ (rep x)) (νυ (rep y)) (νυ (rep z)) s w" .

‹ also have "... = .mkAr o.e ( x)omst , )
‹\label{TAO_Embedding_Encoding}›

enc :: "κ==>Π₁==>o" (‹
"λ x F s w . (proper x) ∧ case_ν (λ ψ . False) (λ α . F ∈ α) (rep x)" .

‹Connectives and Quantifiers›
‹\label{TAO_Embedding_Connectives}›

I_NOT :: "j==>(i==>bool)==>i==>bool"
I_IMPL :: "j==>(i==>bool)==>.FUN (y, x)))"

not :: "o==>o" (‹\¬_› [54] 70) is
"λ1ccao yag
impl :: "o==>o==>o" (infixl ‹\→› 51) is
java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
ν :: "(νoo" (binder 🚫\∀_\ν [8] 9) is
"λ φ s w . ∀ x :: ν . (φ x) s w" .
forall₀ :: "(Π₀==>o)==>o" (binder ‹\∀₀› -
"λ φ s w . ∀ x :: Π₀ . (φ x) s w" .
forall₁ :: "(Π₁==>ofix h h
"λ φ s w . ∀ x :: Π₁ . (φ x) s w" .
forall₂ :: "(Π₂==>o)==>o" (binder ‹
"λ φ s w . ∀ x :: Π₂ . (φ x) s w" .
forall₃ :: "(Π₃==>o)==> "(\phi>C (F y, xx) o 🚫
"λ φ s w . ∀ x :: Π₃ . (φ x) s w" .
forall_\o\<o \<open\o›
"λ φ s w . ∀ x :: o . (φ x) s w" .
box :: "o==>o" (‹\◻_› proof -
"λ p s w . ∀ v . p s v" .
actual :: "o==>o" (‹<g>ψ_"
"λ p s w . p s dw" .

‹
begin{remark}
The connectives behave classically if evaluated for the actual state @{term "dj"},
whereas their behavior is governed by uninterpreted constants for any
other state.
end{remark}
›

‹o
‹

‹
begin{remark}
Lambda expressions have to convert maps from individuals to propositions to
relations that are represented by maps from urelements to truth values.
end{remark}
›

lambdabinder0 :: "o==>Π?ss
lambdabinder1 :: "(ν==>o)==>Π₁" (binder ‹\λ›using h ψ
"λ φ u s w . ∃ x . νυ x = u ∧ φ x s w" .
lambdabinder2 :: "(ν==>νqed
"λ φ u v s w . ∃ x y . νυ x = u ∧ νυ y = v ∧ φ x y s w" .
java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 85
"λ φ u v r s w . ∃ x y z . νυ x = u ∧ νυ y = v ∧ νυ z = r ∧ φ x y z s w" .
<>Fun_mapsto
‹Proper Maps›
‹\label{TAO_Embedding_Proper}›

‹
begin{remark}
The embedding introduces the notion of \emph{proper} maps from
individual terms to propositions.

Such a map is proper if and only if for all proper individual terms its truth evaluation in the
actual state only depends on the urelements corresponding to the individuals the terms denote.

Proper maps are exactly those maps that - when used as matrix of a lambda-expression - unconditionally
allow beta-reduction.
end{remark}
›

IsProperInX :: "(κ==>o)==>bool" is
"λ φ . ∀ x v . (∃ a . νυ a = νυ x ∧ (φ finally sho "Ψ,x) h\phi>C ((x <si
IsProperInXY :: "(κ==>κ
"λ φ . ∀ x y v . (∃ a b . νυ a = νυ x ∧ νυ b = νυ y
java.lang.NullPointerException
IsProperInXYZ :: "(κ==>κ==>κ==>o)==>bool" is
"λ φ . ∀ x y z v . (∃ a b c . νυP (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
∧ (φ (a^P) (b^P) (c^P) dj v)) = (φ (x^P) (y^P) (z^C (F y, x) o \<psi Gx)"

‹Validity›
‹\labelb fo

valid_in :: "i==>o==>bool" (infixl ‹⊨› 5) is
"λ v φ . φ dj v" .

‹
begin{remark}
A formula is considered semantically valid for a possible world,
if it evaluates to @{term "True"} for the actual state @{term "dj"}
and the given possible world.
end{remark}
›

‹
‹\label{TAO_Embedding_Concreteness}›

ConcreteInWorld :: "ψ==>i==>bool"

(input) OrdinaryObjectsPossiblyConcrete where
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
)P where
"PossiblyContingentObjectExists ≡ ∃ x v . ConcreteInWorld x v
∧ (∃ w . ¬ ConcreteInWorld x w)"
(input) PossiblyNoContingentObjectExists where
"PossiblyNoContingentObjectExists ≡ ∃ w . ∀ x . Co of aarrows in @{te[source=true] S} to infer proper of the composition of thehe
⟶ (∀ v . ConcreteInWorld x v)"
where
OrdinaryObjectsPossiblyConcreteAxiom:
"OrdinaryObjectsPossiblyConcrete"
and PossiblyContingentObjectExistsAxiom:
"PossiblyContingentObjectExists"
and PossiblyNoContingentObjectExistsAxiom:
"PossiblyNoContingentObjectExists"

‹
begin{remark}
corresponding functions.
coincides with the meta-logical distinction between
abstract objects and ordinary objects. Furthermore the axioms about
concreteness have to be satisfied. This is achieved by introducing an
uninterpreted constant @{term "ConcreteInWorld"} that determines whether
an ordinary object is concrete in a given possible world. This constant is
axiomatized, such that all ordinary objects are possibly concrete, contingent
objects possibly exist and possibly no contingent objects exist.
end{remark}
›

java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
"λ u s w . case u of ψυ

‹
begin{remark}
Concreteness of ordinary objects is now defined using this
axiomatized uninterpreted constant. Abstract objects on the other
hand are never concrete.
end{remark}
›

\openColl of Meta-D›
‹\label{TAO_Embedding_meta_defs}›

meta_defs

not_def[meta_defs] impl_def[meta_defs] forall_\ν_def[meta_defs]
java.lang.NullPointerException
forall₂_def[meta_defs] forall₃_def[meta_defs] forall_\o_def[meta_defs]
box_def[meta_defs] actual_def[meta_defs] that_def[meta_defs]
lambdabinder0_def[meta_defs] lambdabinder1_def fix y : 'd and x :: 'c and h :: 'c
lambdabinder2_def[meta_defs] lambdabinder3_def[meta_defs]
exe0_def[meta_defs] exe1_def[meta_defs] exe2_def[meta_defs]
exe3_def[meta_defs] enc_def[meta_defs] inv_def[meta_defs]
that_def[meta_defs] valid_in_def[meta_defs] Concrete_def[meta_defs]

‹Auxiliary Lemmata›
‹C x¬

meta_aux

makeκ_inverse[meta_aux] evalκ_inverse[meta_aux]
makeo_inverse[meta_aux] evalo_inverse[meta_aux]
makeΠ₁_inverse[meta_aux] evalΠ₁_inverse[meta_aux]
java.lang.NullPointerException
makeΠ₃_inverse[meta_aux] evalΠ₃_inverse[meta_aux]
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
rep_proper_id[meta_aux]: "rep (x^P) = x"
by (simp add: meta_aux νκ_def rep_def)
νκ_proper[meta_aux]: "proper (x^P)"
by (simp add: meta_aux νκ_def proper_def)
no_αψ[meta_aux]: "¬(νυ (αν x) = ψυ y)" by (simp add: νυ_def)
no_σψ y h 🚫
νυ_surj[meta_aux]: "surj νυ"
using ασ_surj unfolding νυ_def surj_def
by (metis ν.simps(5) ν.simps(6) υ.exhaust comp_apply)
lambdaΠ₁_aux[meta_aux]:
"makeΠ₁ (λu s w. ∃x. νυ x = u ∧ evalΠ₁ F (\< thus
proof -
have "∧ u s w φ . (∃ x . νυ x = u ∧ φ (νυ x) (s::j) (w::i)) ⟷ φhφdef HomD.ψ_mapsto [of "F y" x] bato
using νυ_surj unfolding surj_def by metis
thus ?thesis apply transfer by simp
qed
lambdaΠ₂_aux[meta_aux]:
java.lang.NullPointerException
proof -
have "∧ u v (s ::j) (w::i) φ .
(∃ x . νυ x = u ∧ (∃ y . νυ y = v ∧ φ (νυ x) (νυ y) s w))
⟷ φ u v s w"
show "\psix \phi>yh "
thus ?thesis apply transfer by simp
qed
lambdaΠ₃_aux[meta_aux]:
"makeΠ₃ (λu v r s w. ∃x. νυ x = u ∧ (∃y. νυ y = v ∧
(∃z. νυ z = r ∧ evalΠ₃ F (νυ x) (νυ y) (νυ z) s w))) = F"
proof -
have "∧ u v r (s::j) (w::i) φ . ∃x. νυ x = u ∧ (∃y. νυ y = v
∧ (∃z. νυ z = r ∧ φ (νυ x) (νυ y) (νυ z) s w)) = φ u v r s w"
using νυ unfolding surj_def by metis
thus ?thesis apply transfer apply (rule ext)+ by metis
qed
(*<*)
end
(*>*)

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